Geometric inequality
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A geometric inequality is an inequality involving various measures (angles, lengths, areas, etc.) in geometry.
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Pythagorean Inequality
The Pythagorean Inequality is a generalization of the Pythagorean Theorem. The Theorem states that in a right triangle with sides of length 
we have 
. The Inequality extends this to obtuse and acute triangles. The inequality says:
For an acute triangle with sides of length , 
. For an obtuse triangle with sides , 
.
This inequality is a direct result of the Law of Cosines, although it is also possible to prove without using trigonometry.
Triangle Inequality
The Triangle Inequality says that the sum of the lengths of any two sides of a nondegenerate triangle is greater than the length of the third side. This inequality is particularly useful and shows up frequently on Intermediate level geometry problems. It also provides the basis for the definition of a metric space in analysis.
Isoperimetric Inequality
The Isoperimetric Inequality states that if a figure in the plane has area 
and perimeter 
, then 
. This means that given a perimeter for a plane figure, the circle has the largest area. Conversely, of all plane figures with area , the circle has the least perimeter.
Trigonometric Inequalities
- In
, 
. Proof: 
is a concave function from 
. Therefore we may use Jensen's inequality: 
